This paper by Joel Haddley and Stephen Worsley answers a slightly different question - finding monohedral disc dissections where not all pieces touch the centre - but the results generally apply to this problem too.
My favourite answer is this one:
Note that this one has interior pieces that don't even have a single point on the boundary.
There is an ...
There is a solution which is unique (up to rotation).
If you want to solve it yourself, here is a hint describing the orientation of the pieces:
To be honest, I used a computer to solve it. In general there is no quick way to find a solution to an edge-matching puzzle, or prove whether there is or isn't a solution except in some specific extreme ...